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alright so today I came up with a way to very quickly convert almost any metric value to another

first lets say we have 4678 ml and we want to see how many liters that is

as we all know its 4.678 but bear with me

first I wrote 1000 to represent how many ml in a liter then wrote our number we want to convert on top of it making sure to align the ones place sor now we have

4678
1000

now we make 1000 into 1 to say that we changed that 1000 ml into 1 liter so if we simply place a decimal to make it one we see this 4678 1.000

so now all we have to is bring the decimal up and now we have

4.678
1.000

pretty cool huh? but is it possible to use this trick with non-multiples-of-ten numbers

for instance there are 1024 bytes in a KB so I though...

7000bytes
1024bytes in a kilibytes

add a deciamal, make it one by subtracting the 24 then add the decimal on top and so forth

but this doesn't work

so my question is: is there any way to use the same principles of this little trick to convert other numbers were the number per something isn't a multiple of ten?

any help is appreciated

also heres an example that doesn't work

2.048 (2kb, but we don't know that yet)
1.024
-.024


not 2

thanks

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3 Answers 3

up vote 3 down vote accepted

Yes, but only if you use different bases. For the mL/L example, your technique works because $1$L$ = 10^3$mL, which means you can append $0$'s onto the end of the number of liters to get the number of milliliters. For bytes/kilobytes, $1$kb$ = 2^{10}$b so you can instead use base 2, in which 1kb = 10000000000b (this is base 2 remember, not base 10).

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heh, look at that, it worked... :D thanks –  Luke San Antonio May 2 '11 at 0:57

You are observing the following property of fractions:

$$\frac{a/c}{b/c} = \frac{a}{b}$$

for any nonzero $c$. In the first case you have $a = 4678, b = 1000, c = 1000$ and you got $1$ in the denominator, but in the other cases your denominators are not equal to $1$, so you're getting $\frac{7}{1.024}$ and not $\frac{7}{1}$, for example.

It is true that $\frac{1}{1.024}$ is approximately, but not quite, equal to $1 - 0.024 = 0.976$. In fact, the correct value is $0.9765625$, but the error is small, basically because we are truncating terms in the geometric series

$$\frac{1}{1 + r} = 1 - r + r^2 - r^3 \pm ...$$

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The reason that the trick works for multiples of ten is that moving the decimal point one place left corresponds to a division by ten, and moving it one place to the right corresponds to a division by ten, although I'm sure you knew that. Basically what you're doing is dividing your mL value by 1000 to get your value in liters. Likewise, you'd be dividing by 1024 to get your 7000B into kB, which is accomplished by moving the dot lg(1024)=10 points to the left... in binary. So, if you were to convert 7000 into binary, getting 1101101011000, then you could move the binary point ten points to the left to get 110.1101011000 in base 2 or 6.8359375 in base 10.

As far as I know that's the only similar trick.

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